Consider a flat universe with at least one finite cyclic spatial dimension: travel x meters in one direction, and you will end up back where you started.

For an object that is of small size relative to the scale of the cyclic dimension, relativistic length contraction ought to work out just fine; the cyclic nature of the space doesn't matter, and the object appears contracted in its direction of motion.

For sufficiently large objects, however, there appears to be a paradox. Consider a solid rod of length x, oriented along the cyclic dimension, so that it wraps around the universe and reconnects with itself. Topologically, it's a circle, but everywhere straight and flat. If the rod is accelerated along its length, it should appear to contract; this, however, would make it not long enough to span the cyclic space; thus, one should expect a discontinuity where two ends of the rod will break apart. There is, however, no unique location at which this discontinuity could occur.

So, what's going on? Is a flat cyclic spacetime simply not possible? Or am I missing some deeper understanding of special relativity?


2 Answers 2


Why do you assume the length needed to "span the cyclic space" would be the same in different frames? With cyclic universes it helps to think of them as equivalent to an infinite universe where matter just repeats cyclically, see the "tiling diagram" with the bee and the spider on this page. From this perspective, it's clear the width of a given tile shrinks as well--cyclic universes necessarily involve a globally preferred reference frame where the width is maximum, see this paper for example.

  • $\begingroup$ Because it did not even occur to me that that was an assumption! Once you point it, it suddenly seems obvious. $\endgroup$ Dec 14, 2014 at 0:03

There is no length contraction in circular movements! Length contraction is only in the direction of movement.

A cyclic universe would be e.g. a 3-dimensional universe curved in a fourth dimension, or if a cyclic dimension is curved in a second dimension Thus length contraction could happen only locally, where the curving is not perceivable.


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